Questions tagged [coprime]
Use this tag for questions related to integers such that the only positive integer that divides them is 1.
274
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A Diophantine equation involving divisor counting function?
While working on a different problem, I stumbled upon the following question. Since number theory is not my primary field, I hope somebody with more expertise in number theory can show me a way out.
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Solving $x^a\equiv b \pmod {p^m}$ [duplicate]
I am studying number theory by my own (I'm not a math major but I'm very interested in this, so I'm sorry I don't know much) and I was wondering if there is a result that says if this next equation ...
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Finding Positive Integers for Coprime Arrangements in a Circular Sequence
Find all positive integers $n$ such that the numbers from 1 to $n$ can be arranged in a circle, each appearing exactly once, so that any two numbers between which there is exactly one other number ...
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Why is the sum of cyclic submodules of coprime orders direct? [duplicate]
In theorem 6.4 of Steven Roman's Advanced Linear Algebra, he shows:
If $M$ is a module over a PID $R$ and $u_1, \dots, u_n\in M$ have coprime orders, then the sum
$$\langle\!\langle u_1\rangle\!\...
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Coprime cycle problem [closed]
Problem: a set of integers has a coprime cycle if it can be listed in a cyclic list of the form $(a_1, a_2, a_3, \ldots, a_n)$, where
each element of the set appears exactly once and $\gcd(a_i, a_{i+...
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Maximum consecutive multiples of coprimes [closed]
Given a set of coprimes, can we calculate a value or a limit for the maximum number of consecutive multiples of these values that can possibly occur?
Now, given the same set of coprimes, if each value ...
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2
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1
answer
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Prove that all numbers in the series are relatively prime to each other [duplicate]
I have been working a bit with the following question:
"Prove that all numbers given by $a_n=2^{2^n}+1$ are coprime to
each other"
I think I have a proof but I can't quite finish the last ...
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Are coprimes to the set of primes less than the $i^{th}$ prime always between one multiple of $p_i$?
I am not deeply familiar with mathematics, but I've run into an interesting question while programming that I would appreciate some help with. I'll provide context on the programming side of things to ...
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What is the max value of $|S|$ where $S$ is a set with the following conditions? [duplicate]
$S$ is a subset of the first 1000 integers.
No two elements of $S$ differ by 4 or 7.
I have tried dividing the set of integers upto 1000 into subsets of 4 and 7 integers and tried to include into $S$ ...
1
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1
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My idea about the number of coprime pairs up to $N$.
Today, I wanted to write a program to count how many integer pairs $(a, b)$ that satisfy:
$$1 \leq a < b \leq n, \gcd(a, b) = 1$$
My first instinct was to write a function that check every pair. ...
- 411
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What is $N=1,3,5,9,17,77,105,111,429,455\ldots$, where the second player (possibly) always wins in a game involving incrementing to coprime numbers?
Alice and Bob are playing a game. They take $M$ turns, starting with one point each. The winner is the player with the highest score at the end of the game. In each round the players can make "...
- 1,451
2
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1
answer
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Proportionality of a system of polynomials
I am currently reading Ivanovs „Easy as Pi?“ in fact, I am trying to understand a proof in this book. Following statement is to prove:
For $n\geqslant 3$ the curve $x^n+y^n=1$ has no rational ...
2
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0
answers
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Game strategy of incrementing to coprime numbers with $N=3$: does the second player always have a winning strategy?
Alice and Bob are playing a game. They take $M$ turns, starting with one point each. The winner is the player with the highest score at the end of the game. In each round the players can make "...
- 1,451
3
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1
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What is the remainder when dividing $a$ by $5$ if $\sum_{k=1}^{1992}\frac{1}{k}=\frac{a}{b}$?
Consider
$$\sum_{k=1}^{1992}\frac{1}{k}=\frac{a}{b}.$$
If $a$ and $b$ are natural numbers that are relatively prime, what is the remainder when dividing $a$ by $5$?
$\text{(A) } 0 \space\space\space\...
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Is it possible to prove that if $x$ and $y$ are co-prime, then $(x-y)$ and $\sqrt {xy}$ are also co-prime?
I was trying to prove that pythagorean triplets exists in Natural Number domain.
Here's simplified argument that I did:
consider two natural numbers $x$ and $y$ such that $x > y \ge 1$.
$(x + y)^...
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